There are plenty of terrible jobs in Britain’s febrile political life at the moment, but being leader of the Scottish Labour party must come high on the list. Labour was obliterated by the Scottish National party at the 2015 general election and pushed into third place by the Tories in the election for the Scottish parliament last May. Scotland used to be a Labour fiefdom; now the party is fighting for its life.

The leader faced with the task of saving it is Kezia Dugdale, a personable 34-year-old who critics say is far too nice for such an undertaking. I’ve come to a community centre at the heart of the Old Town in Edinburgh to hear Dugdale give a speech on the lessons of the EU referendum. The premise is a good one: so far, all the attention has been on the majority in Scotland for remain, but what about the million Scots who voted to leave; what were they telling us? As in England, many were poor and felt disempowered; this was a vote about austerity, poverty and powerlessness rather than a straightforward rejection of the EU. Europe was merely a convenient whipping boy.

The promising premise of Dugdale’s speech is undermined by her fondness for political cliche: “Here’s the thing”; “It’s in our DNA”; “Entering uncharted waters”; “It’s a wake-up call”. Such Milibandian language does not bode well. But at a round table with the terriers of the Scottish press immediately afterwards she gives a good account of herself, talking quickly, directly, succinctly.

Dugdale: UK must allow Scottish independence vote if people want it

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When I eventually get her on her own, I ask her why on earth she wanted this terrible job – the political life expectancy of Scottish Labour leaders tends to be short. “I took my time to decide whether I wanted to do it,” she admits – she was Jim Murphy’s deputy when he resigned after the meltdown in May 2015. “If the Labour party were doing extremely well, I wouldn’t be where I am. When Jim quit, I was faced with a fundamental choice: either step up or step away. I chose to step up. I did it through a sense of duty rather than some sort of terrible, overarching careerism.”

She took it on, she says, because she believes Labour can fight back in Scotland. “We’re the only party that stands for the redistribution of both wealth and power.” Hence her emphasis on the people who voted leave – she sees Labour’s role as giving them a stake in a society they feel has abandoned them.

Dugdale’s immediate problem in the wake of the EU referendum is that she and her party may have to choose between being in the UK or the EU. “I find that the most upsetting consequence of the result,” she says. “I am careful about how I talk about identity in politics because identity is a very personal thing. It’s unique to each of us. I feel Scottish first and foremost, but I also feel British and European, and the result of the referendum is that you are asking me to choose between different components of my identity. I don’t want to choose between two unions.”

I, like the vast majority of Scots, want to be part of both the EU and UK

I ask her to try. “We don’t have the full picture of what we’re faced with yet. Nowhere in your life would you make big decisions without being fully versed in the facts of what you are being asked to choose between. We just don’t know whether it’s possible Scotland can remain part of Europe and part of the United Kingdom. I, like the vast majority of Scots, want to be part of both. That’s what I want to fight for.”

She has asked Labour super-fixer Lord Falconer to explore the possibility of a federal solution that would allow Scotland to remain a member of both. “Charlie Falconer is a constitutional expert and, like lots of other people, has a huge desire to maintain both Scotland’s membership of the UK and the European Union. With Ian Murray [Labour’s sole Scottish MP], he is exploring the possibility of a federalised UK option where different federal states can have different relationships with Europe.”

Dugdale backed a motion in the Scottish parliament giving first minister Nicola Sturgeon a mandate to talk to both the UK government and other EU governments about the way ahead for Scotland. Sturgeon has made it clear another referendum on independence is one of the options she is considering. I suggest that makes the breakup of the UK very likely, but Dugdale demurs.

“Before the EU referendum result, Nicola Sturgeon said she would only contemplate a second referendum if the polls consistently showed support [for yes] of over 60%. There have been two polls since the EU referendum, in the midst of this political crisis with the anger that people feel, and they showed 57% and 54% of people saying they would vote yes. I don’t think there’s anything inevitable about a second independence referendum, let alone the inevitability of the country voting yes.”

Dugdale says the UK leaving the EU makes the questions about currency, borders and security that dogged the SNP in the 2014 independence referendum even more pertinent. “We’d have one land mass and potentially two different currencies. What does that mean for trade given that Scotland’s largest trading partner is the rest of the United Kingdom? Is that a good economic step? I struggle to see that it would be.”

She says it would be dangerous for Westminster to block a second independence referendum if that’s what Scots want, but reckons Sturgeon has a tricky hand to play – the Scottish people are unlikely to vote for independence if the byproduct is impoverishment. Falconer’s “federalised” solution or some deus ex machina that sees Brexit averted or diluted is at least as likely as the UK breaking apart.

The other quagmire in which Dugdale finds herself is Jeremy Corbyn’s leadership of the Labour party nationally. She has been sceptical about him from the start, voting for Yvette Cooper in the leadership contest and wondering aloud how a perennial rebel such as Corbyn could enforce party discipline. She said recently that if she had lost the support of 80% of her MPs, she would feel obliged to step down – despite her mandate being even bigger than Corbyn’s, having won 72% of party members’ votes in Scotland in August 2015.

But she has to tread carefully. Her deputy, Alex Rowley, and quite a few Labour members of the Scottish parliament are pro-Corbyn, and Momentum, which sees itself as Corbyn’s praetorian guard, are an increasing presence in Scotland. They are a presence at Dugdale’s Edinburgh speech, too, setting up a stall outside the community centre and handing out leaflets decorated with a photograph of a benign-looking Corbyn standing alongside shadow chancellor John McDonnell, who is giving a clenched-fist salute.

There is, though, less animosity than in England: members of Momentum attend the meeting and make their points (to loud applause) in the Q&A session that follows Dugdale’s speech, and afterwards she chats happily to them, welcoming one new member who says she has joined to protect Corbyn from those seeking to defenestrate him. Dugdale sets great store by being a team player and a democrat – she accepted the Scottish party’s opposition to renewing Trident, even though she personally supports it. That is admirable in its way, but may also have a downside: at some point, a leader has to lead, whatever the cost to party unity.

Dugdale about to speak in the wake of the EU referendum. Photograph: Murdo Macleod/The Guardian

In fairness, she is clear it’s time for Corbyn to go. Pressed by one of the Scottish journalists at the round table, she agrees he is no longer “competent” to run the party because he has lost the support of the great majority of his MPs. “I didn’t actively choose to get the knife out,” she tells me later. “I sought to avoid it for as long as I could. But I know in my heart and in my gut that if I had lost the faith of 80% of my colleagues I could not physically do my job. Like him, I have a mandate from the party membership which I respect and am very conscious of, but the reality is I couldn’t do my job if I’d lost the backing of my colleagues.”

She doubts whether Corbyn has the flexibility to be prime minister; indeed may prefer to be in opposition rather than government, with all the messy compromises it demands. “He’s deeply driven by his principles and wanting to do the right thing,” she says. “He won’t compromise them in order to be in government and he doesn’t think that he needs to. I don’t seek actively to speak ill of him, but I want a Labour government. You can achieve some things from opposition, but nothing like the possibility of power. I get up and do what I do every day because I believe a Labour government could transform this country.”

Dugdale accepts the current infighting is terrible for the party. “Whether you want him to go or stay, everybody in the Labour party just now is hurting. We, as ordinary members, put blood, sweat and tears into this beautiful thing that’s done so much good over decades, and to see it in the position it’s in is upsetting and painful for everyone, regardless of where in the political spectrum you sit.”

She hopes the party doesn’t split, breaking into its socialist and social democrat parts. “The party is a broad church, and that’s what gives it its strength. It has the capacity to represent the whole of the country. It has to speak to crofters in Caithness as much as it does people in Exeter, Truro, Kent and Yorkshire.”

The only child of two teachers who divorced when she was 15, Dugdale grew up in north-east Scotland (in the small town of Elgin and later Dundee), was clever and hard-working – she was head girl at her comprehensive school in Dundee – and went to Aberdeen University to study law. She disliked the course from the start, but stuck with it and was eventually drawn to the ethical questions of the law, though she never intended to practise, which she says distressed her parents.

Instead, she moved to Edinburgh with a group of friends, gathering a large number of rejections to job applications and doing bar work, waitressing and temporary sales jobs to tide her over. That experience led her to join the Labour party. “There were three of us who’d moved down from Aberdeen, and only one of my flatmates got a job. My other friend couldn’t get a job either, and we would sit on the couch all day chewing the fat. She was a member of the Labour party, and she said: ‘All these things that you think, that’s what the Labour party believes.’”

Dugdale joined, got a job working for Edinburgh University as a welfare adviser, did a part-time MA in social policy, and in 2007 became an election agent in Edinburgh. “I love elections,” she says. “I love the theatre of elections, the geekery of it, and the more I got involved, the more I wanted to do.” She went to work for then Labour MSP George Foulkes, enjoyed the cut-and-thrust of life in the Scottish parliament, and became an MSP herself in 2011.

Doesn’t all this leave her open to accusations of wonkery? “A lot of people look at my age and assume I decided I was going to be a politician when I was 12,” she says. “That’s just not the case. I didn’t vote till I was 23; I probably joined the Labour party before I voted. I’ve worked in countless cafes and bars, stacked oil rigs with goods, worked in numerous call centres. I don’t like anyone being accused of being a careerist because all they are actually doing is trying to live by their values and deliver what they believe in.”

Dugdale came out just before the Scottish election in May – she has been with her partner, economics lecturer Louise Riddell, for more than eight years. Why did she choose to go public about their relationship? “If you speak to journalists here in Scotland, everybody knew. I had just chosen not to talk about it because my partner gave me the stability and security I need to do my job. It was the one bit of my life I needed to be mine. But I got tired of not talking about it, and since that point I’ve been trying to make up for lost time by being an advocate for young LGBT people.”

To me, as an outsider, the way politics is conducted in Scotland feels very different from down south: more civilised and harmonious, a bit quirkier too. Dugdale’s father, Jeff, is an SNP supporter who is not averse to having a pop at his daughter on Twitter (she calls him a typical “embarrassing dad”). I suggest this different political mood reflects the dominance of women in political life in Scotland – all three main parties are led by women, two of whom are gay – but Dugdale says that’s too pat an explanation.

We’re just as robust in how we do our politics as three men would be

“I think it is different, but only a little bit different,” she says. “People will often say: ‘It’s three women, so it must be less aggressive.’ But they’ve clearly not been watching because we’re just as robust in how we do our politics as three men would be.” She also says the family friendliness of Scottish politics is overstated, pointing out that none of the leaders of the three main parties has children, though she and her partner hope to have them at some point.

Her task now is to steer Scottish politics off the constitutional agenda and on to the social and economic agenda that she believes favours Labour. It has been difficult for Labour to regain lost ground in a framework where the SNP represents nationalism and the Conservatives seek to embody unionism – falsely, in her view, as the decision of a Conservative government to hold a referendum on the EU has imperilled the very union the Tory party seeks to uphold.

She says referendums reduce political decisions to the lowest common denominator, demanding simple answers to complex questions, and herding wildly different groups of people into the same ideological pen. “Over the past three or four years, we have become obsessed by binary choices. You’re either yes or no, for or against something, black or white, leave or remain. Why are there only ever two options now in our politics? What happened to the grey areas? That’s where consensus and progress are found.”

In praise of grey areas – it’s admirable in its way, but you do fear she might be swimming against an unstoppable tide of ideological purity on left and right.